The Information Bottleneck's Ordinary Differential Equation: First-Order Root Tracking for the Information Bottleneck

Area of Science:

• Information Theory
• Machine Learning
• Signal Processing

Background:

• The Information Bottleneck (IB) method offers lossy compression of relevant information.
• Its rate-distortion (RD) curve illustrates the trade-off between input compression and information preservation.
• Existing methods obscure the dynamics of optimal input encodings.

Purpose of the Study:

• To reveal the underlying dynamics of optimal input encodings in the Information Bottleneck.
• To address the limited numerical techniques for solving IB problems.
• To leverage IB-RD relationships for improved solution structures and numerical algorithms.

Main Methods:

• Analyzing the piecewise smooth trajectory of optimal encodings.
• Identifying and classifying bifurcations where encoding dynamics change qualitatively.
• Deriving the Information Bottleneck's first-order Ordinary Differential Equation (ODE).
• Developing a numerical algorithm that exploits ODE dynamics and handles bifurcations.

Main Results:

• Optimal input encodings exhibit piecewise smooth dynamics interrupted by bifurcations.
• Sub-optimal solutions can collide or exchange optimality at bifurcations.
• A novel ODE accurately describes the dynamics of the IB's optimal tradeoff curve.
• The proposed numerical algorithm accurately follows the optimal tradeoff curve by handling bifurcations.

Conclusions:

• Understanding IB bifurcations provides crucial insights into solution structures.
• The derived ODE and bifurcation analysis enable a more accurate numerical solution for IB problems.
• This work offers a more efficient and accurate approach to solving Information Bottleneck problems.